Problem 65 - An Introduction to Mathematical Statistics and Its Applications

Question

In a certain country, the distribution of a family's disposable income, \(Y\), is described by the pdf \(f_{Y}(y)=\) \(y e^{-y}, y \geq 0\). Find \(F_{Y}(y)\).

Step-by-Step Solution

Answer

The cumulative distribution function \(F_{Y}(y)\) for the given probability density function is \(F_{Y}(y)=1 - (x+1)e^{-x}\).

Step 1: Identification of the given pdf

The task provides the pdf as \(f_{Y}(y)=y e^{-y}, y \geq 0\). This is our starting point for calculating the CDF \(F_{Y}(y)\).

Step 2: Definition of CDF

The cumulative distribution function \(F_{Y}(y)\) for a random variable \(Y\) with a pdf \(f_{Y}(y)\) is defined by the integral from \(-\infty\) to \(y\) of \(f_{Y}(x)\) dx. However, as our pdf is valid only for \(y \geq 0\), the lower limit of the integral will be 0.

Step 3: Calculation of CDF

Now we calculate the CDF by integrating the pdf from 0 to \(y\). This gives us \(F_{Y}(y)=\int_{0}^{y}x e^{-x} dx\). Integrating by parts using the formula \(\int u dv = uv - \int v du\), where \(u=x, dv=e^{-x} dx\), we find that \(du=dx\) and \(v=-e^{-x}\). This gives us \(-x e^{-x} - \int_{0}^{y}-e^{-x} dx\) which simplifies to \(-x e^{-x} + \int_{0}^{y}e^{-x} dx\). Integrating \(e^{-x}\) from 0 to \(y\) we get \(-e^{-x}\). Therefore, our final result is \(-x e^{-x} - (-e^{-x} -1)\), which simplifies to \(1 - (x+1)e^{-x}\).

Key Concepts

Probability Density Function

A probability density function, commonly abbreviated as **pdf**, is a fundamental concept in statistics and probability theory. It represents how probabilities are distributed over the values of a random variable. If you imagine a curve, the height of the pdf at a given point represents the likelihood of the random variable taking on that specific value. This is defined for continuous random variables.
Here are some key points about probability density functions:

• **Non-Negative Values**: A pdf is always non-negative, meaning it cannot be less than zero at any given point.
• **Integration Equals One**: The integral of the pdf over the entire range of possible outcomes must equal one, which ensures the total probability is 100%.
• **Describes Likelihoods**: Rather than giving exact probabilities, a pdf shows the likelihood of the variable falling within a specific range.

In our exercise, the given pdf is \( f_{Y}(y) = y e^{-y} \). Here, \( y \) is the variable representing the family's disposable income, and this pdf is valid for all \( y \geq 0 \). The pdf is an essential tool in determining the cumulative distribution function (CDF), which provides the probability that the random variable is less than or equal to a certain value.

Integration by Parts

Integration by parts is an important technique used in calculus to integrate products of functions. This method is especially useful when traditional integration tactics are difficult to apply. It's derived from the product rule for differentiation and is expressed by the formula:\[\int u \, dv = uv - \int v \, du\]This formula requires selecting two functions to differentiate and integrate, respectively:

• **Choose \( u \)**: Often choose \( u \) to be the function labeled as more "difficult" to differentiate.
• **Choose \( dv \)**: Select the remaining part as \( dv \) to simplify following steps.

In our solution, integration by parts helps handle the integral \( \int x e^{-x} \, dx \). Here, we choose \( u = x \) and \( dv = e^{-x} \, dx \). Upon applying integration by parts, this original integral breaks down into parts that are simpler to evaluate, ultimately helping us find the cumulative distribution function.

Exponential Distribution

The exponential distribution is a commonly used probability distribution in statistics, particularly useful in predicting the time until an event occurs. It is a special case of the gamma distribution and has a distinct probability density function.

• **Memoryless Property**: One of the key features of an exponential distribution is its memoryless property. This means that the probability of an event occurring in the future is independent of any past events.
• **Parameters**: It is generally parameterized by a rate \( \lambda \), which is the inverse of the mean of the distribution.
• **Shape**: The pdf of an exponential distribution generally decreases steadily, reflecting the decreasing likelihood of a longer waiting time.

Although the given pdf in our exercise \( f_{Y}(y) = y e^{-y} \) is not a pure exponential distribution, the term \( e^{-y} \) suggests an exponential nature in part of the formula. Understanding how exponential distributions behave can provide insights into the patterns and trends suggested by this pdf.